Simulation of Sound Spectra of Complex Didgeridoo Interior Forms
Computer Aided Didge Sound Design

Dr. Frank Geipel


Frequenzmuster im Didgeridoo

Motivation
Every advanced didgeridoo player will have found that the sound characteristics (toots, timbre, ...) and the playability (back-pressure, responsiveness…) are essentially influenced by the inside form and the material of the instrument. I know many players who are forever searching for a didgeridoo with their ideal sound and playability because of this. Often it’s a very sobering search because very few of the instruments offered on the didgeridoo “market” fully meet one’s expectations. And many of those one-offs that do are also very expensive.
 

This situation prompted me to get into building didgeridoos. I soon found that thorough preparation could put me more than half-way to my goal and spare me many failed building experiments.


The way to the own ideal didge

Frank prüft die Leimfuge eines neuen Didges - sie ist nicht mehr zu sehen ...
1) Building methods

In the didge scene there are opportunities to thoroughly inform yourself about the crafting skills needed to make the instrument. Because I wanted to work with wood, I had the choice between the sandwich method (practised by Eddy Halat, Stefan Thiel, Jan-Ole Haber, Kay Reimer, and others) and the drilling method (practised by Walter Strasser, Johannes Schildkamp and others in several variations). Unfortunately I’ve yet to hear of “trained termites” in central Europe.
Interesting though the drilling method practised by a few craft artists is, I opted for the simpler sandwich method because it allows considerably more complex internal forms to be realised.
While I was researching the methods I got to know some interesting didgeridoo makers, including the Test-A-Doo experimenter, Kay Reimer, whose acoustic experiments impressed me very deeply. I followed his web instructions to make my own first wooden didgeridoo.
 

Sehr gelungen - Franks letztes Didge!

Birch wood didgeridoo fundamental tone F# first overblow A2

2) Using wood

In regard to material properties, I prefer hardwood varieties of high density and high elasticity (elasticity modules). These are the closest to eucalyptus and least restrict the higher overtones. For the selection of suitable wood varieties I have compiled some interesting wood data on our website which can be very helpful to gauge the resonance qualities and the ease of working.
Specifically, I use e.g. plum/damson, hawthorn (tends unfortunately to split), hop hornbeam, hornbeam, oak, yew (heavy, practically never splits, high elasticity), ash, robinia and birch (medium hard and medium heavy but very high elasticity).


3) Bore shapes

Frank beim Grübeln - irgendie muss die Simulation zu schaffen sein! How do you achieve suitable didgeridoo interior shapes, which provide the desired sound characteristics? Here each didgeridoo craftsman has made his own experiences, which of course commercially motivated craftsmen will only rarely share. As an interested scientist, I had the idea of finding such interior shapes through computer simulation, hence avoiding a multitude of unsuccessful building experiments.

 

Unfortunately research on this topic only provides suitable mathematical solutions for simple cylindrical and exactly conical tubing shapes (formula sides 1 and 2).



I was not very interested in these simple, idealized forms, because the sound characteristics that can be generated in this way are very limited and are not correct in the case of deviations from these idealized forms.
 

I had to find another method!

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The method 

I searched for a method by which the passive acoustic properties of more complex didgeridoo internal forms are calculable.
 

 

That aim in mind, I started a private project in early 2003. After extensive research in scientific literature and discussions with physicists, I opted for the method of transmission line modelling, which I developed further with my own ideas and findings.

 

With this method, a didgeridoo is mathematically split into a finite number of cylindrical and conical pieces. The acoustic chain matrix in the complex numbers sector can be resolved for thus modelled inner cavities of didgeridoos, taking into account the uneven inside walls. In this way one obtains the so-called input impedance spectra, from which the resonant frequencies of the air column and the back-pressure of the drone (fundamental tone) and toots (series of overblows) can be read. By coupling the impedance spectra with the respective simulated overtone spectra when playing the drone or toots, one obtains additionally the sound spectra for the drone and the first toot. These simulated sound spectra match well with the practically analysable FFT (Fast Fourier Transformation) spectrograms when the individual instruments are played.

In the context of this website I will predominantly work with absolute frequency data. In order to be able to achieve an allocation to the musical tone designations at any time, the respective frequencies are assigned to the tones in table 1.

Table 1

Audio frequencies in Hz; The range marked in grey shows the fundamental tone frequencies in which most didgeridoos resonate.
 

The following example shows the sound simulation of actual measurements of one of Walter Strasser’s didgeridoos and the accompanying practical analysis of the FFT spectrogram with an FFT analysis programme.

Walter Strasser’s didgeridoo; black: simulation of the toot sequences (drone and overblows);
grey: Simulation of the sound spectrum when the drone is played;
(L=135 cm, dMouth=30 mm, dBell=110 mm; Drone E1, first toot A3+ ) characterised by “singing” third overtone at E3 louder than the drone E11

Practical analysis of the FFT spectrum when playing the drone; the sound level depicted is a relative logarithmic value expressing the difference between acoustic pressures. However, to interpret these spectra the subjective perception of volume of the concurrently sounding frequencies is important. Without wanting to delve any deeper into that aspect, it can be assumed that from the highest peak (i.e. the loudest frequency) only those frequencies still significantly influence the sound character which are up to about 40 dB below the highest peak. All quieter frequencies are overlaid by the louder ones. That means that for every FFT spectrum there exists a sound level section (green box=, that significantly determines the sound characteristic.

Wave pattern for this didgeridoo

Current outcomes of the project are various prototype software tools by which, in dependence of complex internal forms, the toot sequences (drone and overblows) and the sound spectra of the drone and 1st toot can be simulated/calculated. Presently these tools are to hand as non-self-explanatory work versions still under development.

The following example shows the sound simulation of a very interesting interior form with parallel amplified 4th and 5th harmonic overtone. The first overblow is located an oktave and a tone over the fundamental tone and should be very easily to play. The building of a Didgeridoo with this sound characteristic by suitable software produced templates is documented in the crafting example.

Simulation of a didgeridoo with parallel amplified 4th and 5th harmonic overtone
- sound spectrum of the fundamental drone: dark green
- sound spectrum of the 1st overblow: light green
- series of overblows / resonances: white

Using the method developed opens up the following possibilities:

1) Didgeridoos with complex internal shapes can be projected in such a way that the drone and the playable toot sequences can be predetermined.

2) So-called “singing” didgeridoos can be made in which one or two desired overtones are strengthened by higher acoustic impedance peaks in the range of e.g. 350-750 Hz. In the scene such didgeridoos are usually rarities.

3) Didgeridoos can be modelled which have pronounced acoustic impedance peaks between the first harmonic overtones. In these frequencies the voice is especially strengthened and is easier to use in an accentuated manner.

4) FFT spectra can be recorded of interesting didge sound characteristics available as recordings to model internal forms that can come very close to the desired sound characteristics. Due to the fact that the internal forms essentially determine the passive sound characteristics, in principle these internal forms are reconstructable from the sound spectra analysed.

5) In a relatively short time the sound characteristics of so many different internal forms can be simulated that are practically impossible to realise with classical crafting methods. Despite that, however, the know-how of experienced didgeridoo craftsmen is needed to practically implement the simulated internal forms optimally.

The simulation of sound spectra of many hundreds of didgeridoo interiors with a computer makes clear that just small changes of the inner form can open up a “universe” of possible sound spectra. The ultimate didgeridoo that realises all possible wishes about sound characteristics is not calculable. What is possible, though, is an almost endless variety of didgeridoos with unique sound and playing qualities.
 

Frequenzmuster im Didgeridoo

For anyone with a deeper interest in this theme, I can recommend the book “Das Didgeridoo-Phänomen”. The chapter “Simulation of sound spectra of complex internal didgeridoo forms – Computer Aided Didge (Sound) Design” describes in detail the methodology, the physical correlations and the possibilities for didgeridoo building.

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